Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

185 submissions , 145 unreviewed
4,725 questions , 1,928 unanswered
5,268 answers , 22,442 comments
1,470 users with positive rep
743 active unimported users
More ...

  Thought experiment in relativistic quantum mechanics?

+ 1 like - 0 dislike
112 views

Background
---
Consider the following thought experiment in the setting of relativistic quantum mechanics (not QFT). I have a particle in superposition of the position basis:

$$ H | \psi \rangle = E | \psi \rangle$$

Now I suddenly turn on an interaction potential $H_{int}$ localized at $r_o = (x_o,y_o,z_o)$ at time $t_o$:

$$
H_{int}(r) = 
\begin{cases} 
      k & r \leq r_r' \\
      0 & r > r' 
   \end{cases}
$$

where $r$ is the radial coordinate and $r'$ is the radius of the interaction of the potential with origin $(x_o,y_o,z_o)$

By the logic of the sudden approximation out state has not had enough time to react. Thus the increase in average energy is:

$$ \langle \Delta E \rangle = 4 \pi k \int_0^{r'} |\psi(r,\theta,\phi)|^2 d r  $$

(assuming radial symmetry).

Now, lets say while the potential is turned on at $t_0$ I also perform a measurement of energy at time $t_1$ outside a region of space with a measuring apparatus at some other region $ (x_1,y_1,z_1)$. Using some geometry it can be shown I choose $t_1 > t_0 + r'/c$ such that: 

$$ c^2(t_1 - t_0 - r'/c)^2 -(x_1 - x_0)^2  - (y_1 - y_0)^2 - (z_1 - z_0)^2 < 0 $$

Hence, they are space-like separated. This means  I could have one observer who first sees me turn on the potential $H_{int}$ and measure a bump in energy $\langle \Delta E \rangle $ but I could also have an observer who sees me first measure energy and then turn on the interaction potential.

Obviously the second observer will observe something different.

Question
---
How does relativistic quantum mechanics deal with this paradox?

asked Jan 16 in Theoretical Physics by Asaint (80 points) [ revision history ]

QM has nothing to do with your behavior.

Direct (retarded) observation results in the relativistic mechanics are recalculated to the proper reference frame. For example, even in Classical mechanics, when you compare the lengths of two equal rods, one of which is next to you and the other is far away (and thus looks "small"), you conclude that the distant rod is of the same length - by recalculating its apparent length to your proper RF.

@Vladimir in the paradox since they are spacelike separated. Observer 2 will see the  measurement first and then the change in potential and conclude that the experimenter will observe $\langle H \rangle$ whereas the first Observer will see the change in potential first and then the measurement and conclude the experimenter has observed $\langle H + H_{int} \rangle$. Both can obviously not be correct.

Two spacelike separated events may be observed as simultaneous in one RF ($\Delta t=0$) and time-separated in another; the sequence of events depending on particular RF position.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ys$\varnothing$csOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...